The quadratic equation $2x^{2} - (a^{3} + 8a - 1)x + a^{2} - 4a = 0$ has roots of opposite signs. Then,

For what values of $a \in \mathbb{Z}$,the quadratic expression $(x+a)(x+1991)+1$ can be factorised as $(x+b)(x+c)$,where $b, c \in \mathbb{Z}$?

If $1+\sqrt{2}$ and $2-i$ are the roots of the equation $x^4+bx^3+cx^2+dx+e=0$ where $b, c, d, e$ are rational numbers,then the roots of the equation $bx^2+cx+d=0$ are

$A$ two-digit number is four times the sum of its digits and three times the product of its digits. The number is:

Let $N$ be the number of quadratic equations of the form $ax^2 + bx + c = 0$ with coefficients $a, b, c \in \{0, 1, 2, \dots, 9\}$ such that $0$ is a solution of each equation. Then the value of $N$ is

The number of solutions for the equation $x^2-5|x|+6=0$ is .........